Implicit solutions with consistent additive and multiplicative components

P. Areias, T. Rabczuk, D. Dias-Da-Costa, E. B. Pires

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

This work describes an algorithm and corresponding software for incorporating general nonlinear multiple-point equality constraints in a implicit sparse direct solver. It is shown that direct addressing of sparse matrices is possible in general circumstances, circumventing the traditional linear or binary search for introducing (generalized) constituents to a sparse matrix. Nested and arbitrarily interconnected multiple-point constraints are introduced by processing of multiplicative constituents with a built-in topological ordering of the resulting directed graph. A classification of discretization methods is performed and some re-classified problems are described and solved under this proposed perspective. The dependence relations between solution methods, algorithms and constituents becomes apparent. Fracture algorithms can be naturally casted in this framework. Solutions based on control equations are also directly incorporated as equality constraints. We show that arbitrary constituents can be used as long as the resulting directed graph is acyclic. It is also shown that graph partitions and orderings should be performed in the innermost part of the algorithm, a fact with some peculiar consequences. The core of our implicit code is described, specifically new algorithms for direct access of sparse matrices (by means of the clique structure) and general constituent processing. It is demonstrated that the graph structure of the second derivatives of the equality constraints are cliques (or pseudo-elements) and are naturally included as such. A complete algorithm is presented which allows a complete automation of equality constraints, avoiding the need of pre-sorting. Verification applications in four distinct areas are shown: single and multiple rigid body dynamics, solution control and computational fracture.

Original languageEnglish
Pages (from-to)15-31
Number of pages17
JournalFinite Elements in Analysis and Design
Volume57
DOIs
Publication statusPublished - 2012 Sep

Keywords

  • Computational fracture
  • Multibody dynamics
  • Multiple-point constraints
  • Sparse matrices

ASJC Scopus subject areas

  • Analysis
  • Engineering(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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